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23. Group Actions and Automorphisms Recall the Definition of an Action

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23. Group Actions and Automorphisms Recall the Definition of an Action 23. Group actions and automorphisms Recall the definition of an action: Definition 23.1. Let G be a group and let S be a set. An action of G on S is a function G × S −! S denoted by (g; s) −! g · s; such that e · s = s and (gh) · s = g · (h · s) In fact, an action of G on a set S is equivalent to a group homomor- phism (invariably called a representation) ρ: G −! A(S): Given an action G × S −! S, define a group homomorphism ρ: G −! A(S) by the rule ρ(g) = σ : S −! S; where σ(s) = g · s. Vice-versa, given a representation (that is, a group homomorphism) ρ: G −! A(S); define an action G · S −! S by the rule g · s = ρ(g)(s): It is left as an exercise for the reader to check all of the details. The only sensible way to understand any group is let it act on some- thing. Definition-Lemma 23.2. Suppose the group G acts on the set S. Define an equivalence relation ∼ on S by the rule s ∼ t if and only if g · s = t for some g 2 G. The equivalence classes of this action are called orbits. The action is said to be transitive if there is only one orbit (neces- sarily the whole of S). Proof. Given s 2 S note that e · s = s, so that s ∼ s and ∼ is reflexive. If s and t 2 S and s ∼ t then we may find g 2 G such that t = g · s. But then s = g−1 · t so that t ∼ s and ∼ is symmetric. If r, s and t 2 S and r ∼ s, s ∼ t then we may find g and h 2 G such that s = g · r and t = h · s. In this case t = h · s = h · (g · r) = (hg) · r; so that t ∼ r and ∼ is transitive. 1 Definition-Lemma 23.3. Suppose the group G acts on the set S. Given s 2 S the subset H = f g 2 G j g · s = s g; is called the stabiliser of s 2 S. H is a subgroup of G. Proof. H is non-empty as it contains the identity. Suppose that g and h 2 H. Then (gh) · s = g · (h · s) = g · s = s: Thus gh 2 H, H is closed under multiplication and so H is a subgroup of G. Example 23.4. Let G be a group and let H be a subgroup. Let S be the set of all left cosets of H in G. Define an action of G on S, G × S −! S as follows. Given gH 2 S and g0 2 G, set g0 · (gH) = (g0g)H: It is easy to check that this action is well-defined. Clearly there is only one orbit and the stabiliser of the trivial left coset H is H itself. Lemma 23.5. Let G be a group acting transitively on a set S and let H be the stabiliser of a point s 2 S. Let L be the set of left cosets of H in G. Then there is an isomorphism of actions (where isomorphism is defined in the obvious way) of G acting on S and G acting on L, as in (23.4). In particular jGj jSj = : jHj Proof. Define a map f : L −! S by sending the left coset gH to the element g ·s. We first have to check that f is well-defined. Suppose that gH = g0H. Then g0 = gh, for some h 2 H. But then g0 · s = (gh) · s = g · (h · s) = g · s: Thus f is indeed well-defined. f is clearly surjective as the action of G is transitive. Suppose that f(gH) = f(gH). Then gS = g0s. In this case h = g−1g0 stabilises s, so that g−1g0 2 H. But then g and g0 are 2 in the same left coset and gH = g0H. Thus f is injective as well as surjective, and the result follows. Given a group G and an element g 2 G recall the centraliser of g in G is Cg = f h 2 G j hg = gh g: The centre of G is then Z(G) = f h 2 H j gh = hg g; the set of elements which commute with everything; the centre is the intersection of the centralisers. Lemma 23.6 (The class equation). Let G be a group. The cardinality of the conjugacy class containing g 2 G is the index of the centraliser, [G : Cg]. Further X jGj = jZ(G)j + [G : Cg]; [G:Cg]>1 where the second sum run over those conjugacy classes with more than one element. Proof. Let G act on itself by conjugation. Then the orbits are the conjugacy classes. If g 2 then the stabiliser of g is nothing more than the centraliser. Thus the cardinality of the conjugacy class containing g is [G : Cg] by (23.3). If g 2 G is in the centre of G then the conjugacy class containing G has only one element, and vice-versa. As G is a disjoint union of its conjugacy classes, we get the second equation. Lemma 23.7. If G is a p-group then the centre of G is a non-trivial subgroup of G. In particular G is simple if and only if the order of G is p. Proof. Consider the class equation X jGj = jZ(G)j + [G : Cg]: [G:Cg]>1 The first and last terms are divisible by p and so the order of the centre of G is divisible by p. In particular the centre is a non-trivial subgroup. If G is not abelian then the centre is a proper normal subgroup and G is not simple. If G is abelian then G is simple if and only if its order is p. Theorem 23.8. Let G be a finite group whose order is divisible by a prime p. Then G contains at least one Sylow p-subgroup. 3 Proof. Suppose that n = pkm, where m is coprime to p. Let S be the set of subsets of G of cardinality pk. Then the cardi- nality of S is given by a binomial n pkm(pkm − 1)(pkm − 2) ::: (pkm − pk + 1) = pk pk(pk − 1) ::: 1 Note that for every term in the numerator that is divisible by a power of p, we can match this term in the denominator which is also divisible by the same power of p. In particular the cardinality of S is coprime to p. Now let G act on S by left translation, G × S −! S where (g; P ) −! gP: Then S is breaks up into orbits. As the cardinality is coprime to p, it follows that there is an orbit whose cardinality is coprime to p. Suppose that X belongs to this orbit. Pick g 2 X and let P = g−1X. Then P contains the identity. Let H be the stabiliser of P . Then H ⊂ P , since h · e 2 P . On the other hand, [G : H] is coprime to p, so that the order of H is divisible by pk. It follows that H = P . But then P is a Sylow p-subgroup. Question 23.9. What is the automorphism group of Sn? Definition-Lemma 23.10. Let G be a group. If a 2 G then conjugation by G is an automorphism σa of G, called an inner automorphism of G. The group G0 of all inner automor- phisms is isomorphic to G=Z, where Z is the centre. G0 is a normal subgroup of Aut(G) the group of all automorphisms and the quotient is called the outer automorphism group of G. Proof. There is a natural map ρ: G −! Aut(G); whose image is G0. The kernel is isomorphic to the centre and so G0 ' G=Z; by the first Isomorphism theorem. It follows that G0 ⊂ Aut(G) is a subgroup. Suppose that φ: G −! G is any automorphism of G.I claim that −1 φσaφ = σφ(a): 4 Since both sides are functions from G to G it suffices to check they do the same thing to any element g 2 G. −1 −1 −1 φσaφ (g) = φ(aφ (g)a ) = φ(a)gφ(a)−1 = σφ(a)(g): 0 Thus G is normal in Aut(G). Lemma 23.11. The centre of Sn is trivial unless n = 2. Proof. Easy check. Theorem 23.12. The outer automorphism group of Sn is trivial unless n = 6 when it is isomorphic to Z2. Lemma 23.13. If φ: Sn −! Sn is an automorphism of Sn which sends a transposition to a transposition then φ is an inner automorphism. Proof. Since any automorphism permutes the conjugacy classes, φ sends transpositions to transpositions. Suppose that φ(1; 2) = (i; j). Let a = (1; i)(2; j). Then σa(i; j) = (1; 2) and so σaφ fixes (1; 2). It is ob- viously enough to show that σaφ is an inner automorphism. Replacing φ by σaφ we may assume φ fixes (1; 2). Now consider τ = φ(2; 3). By assumption τ is a transposition. Since (1; 2) and (2; 3) both move 2, τ must either move 1 or 2. Suppose it moves 1. Let a = (1; 2). Then σaφ still fixes (1; 2) and σaτ moves 2. Replacing φ by σaφ we may assume τ = (2; i), for some i. Let a = (3; i).
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